# What is Indefinite integral: Definition + 207 Threads

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.

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1. ### Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ Which leads...
2. ### How Does Substitution Affect Double Integration and Differentiation?

\frac{d^{2}}{dx^{2}}\int_{0}^{x}(\int_{1}^{sint}\sqrt{1+u^{4}}du)dt=\frac{d}{dx}\int_{0}^{sinx}(\sqrt{1+u^{4}})du then we let m=sinx，so x=arcsinx，then we get \frac{d}{dx}\int_{0}^{sinx}(\sqrt{1+u^{4}})du=\frac{dm}{dx}\frac{d}{dm}\int_{0}^{m}(\sqrt{1+u^{4}})du=\sqrt{1+m^{4}}\frac{dm}{dx}，then we...
3. ### Can't Find a Correct Method to Integrate \int (t - 2)^2\sqrt{t}\,dt?

When I encountereD this kind of question before.For example \int x\sqrt{2+x^{2}}dx We make the Substitution t=x^{2}+2，because its differential is dt=2xdx，so we get \int x\sqrt{2+x^{2}}=1/2\int\sqrt{t}dt，then we can get the answer easily But the question，it seems that I can't use the way to...
4. ### Find the indefinite integral of the given problem

Now the steps to solution are clear to me...My interest is on the constant that was factored out i.e ##\frac{2}{\sqrt 3}##... the steps that were followed are; They multiplied each term by ##\dfrac{2}{\sqrt 3}## to realize, ##\dfrac{2}{\sqrt 3}\int \dfrac{dx}{\left[\dfrac{2}{\sqrt...
5. ### MHB Indefinite integral in division form

I have the following integration - $$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx$$ To solve this I did the following - $$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx$$ Which gives me -...
6. ### The integral of a function ##f(x)## from its graph

Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##. Attempt : I argue for or against each statement by putting it down first in blue and my answer in red. ##g(x)## is always positive : The exact...
7. ### I What is the indefinite integral of Bessel function of 1 order (first k

Hi When we find integrals of Bessel function we use recurrence relations. But this requires that we have the variable X raised to some power and multiplied with the function . But how about when we have Bessel function of first order and without multiplication? How should we integrate it ?

47. ### MHB Evaluation of Indefinite Integral

Calculation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ **My Try \(\displaystyle :: I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin x\cdot \sqrt{\cos 2x}}\cdot \frac{\sin x}{\sin x}dx = \int\frac{\sqrt{2\cos^2 x-1}}{\left(1-\cos^2 x\right)\cdot \sqrt{2\cos^2...
48. ### What is the indefinite integral of cosecant function?

What is the indefinite integral of cosec(\theta)?
49. ### Indefinite Integrals of Scalar and Vector Fields: A Path Independence Dilemma?

Is possible to compute indefinite integrals of functions wrt its variables, but is possible to compute indefinite integrals of scalar fields and vector fields wrt line, area, surface and volume?
50. ### MHB Help with Solving Indefinite Integral

Hi, I tried to solve this integral \int\sqrt{1-\frac{1}{x^3}}dx but i can't solve it... can someone help me?